Definition:Module/Left and Right Modules

Note

In the case of a commutative ring, the difference between a left module and a right module is little more than a notational difference. See:

• Left Module over Commutative Ring induces Right Module
• Right Module over Commutative Ring induces Left Module.

But this is not the case for a ring that is not commutative. From:

• Left Module Does Not Necessarily Induce Right Module over Ring
• Right Module Does Not Necessarily Induce Left Module over Ring

it is known that it is not sufficient to simply reverse the scalar multiplication to get a module of the other ‘side’.

From:

• Left Module over Ring Induces Right Module over Opposite Ring
• Right Module over Ring Induces Left Module over Opposite Ring

to obtain a module of the other ‘side’ it is, in general, also necessary to reverse the product of the ring.

From:

• Left Module induces Right Module over same Ring iff Actions are Commutative
• Right Module induces Left Module over same Ring iff Actions are Commutative

a left module induces a right module and vice-versa if and only if actions are commutative.

Also see

• Definition:Left Module
• Definition:Right Module
• Definition:Module
• Definition:Scalar Ring of Module
• Basic Results about Modules

• Results about modules can be found here.